Defining the normal distribution

The normal (also known as ‘Gaussian’) is \(X\sim N(\mu,\sigma^2)\):

• \(\mu\) is the expected value (\(E[X]\))
• \(\sigma^2\) is the variance of (\(Var[X]\))

Its pdf:

\[ f(x)=\frac1{(2\pi\sigma^2)^{1/2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

(Just like the binomial, you don’t need to know this formula for this class)

Computing probabilities: old vs. new

Computing probabilities with the normal requires computers

pnorm(1.8,0,1)

[1] 0.9640697

pnorm(1.89,0,1)

[1] 0.970621

Computing probabilities

We will always write \(X\sim N(\mu,\sigma^2)\) for the normal

R syntax will be specified with \(\mu\) and \(\sigma = \sqrt{\sigma^2}\) instead

Example

If \(X \sim N(5,9)\), then the probability \(X\) is less than 4 is:

x       = 4
mu      = 5
sigmaSq = 9
pnorm(x, mu, sqrt(sigmaSq))

[1] 0.3694413

Computing values (quantiles)

Continuing with \(X \sim N(5,9)\):

(that is, normal with expected value 5 and variance 9)

pnorm(4,5,3)

[1] 0.3694413

qnorm(.7,5,3)

[1] 6.573202

Using pnorm and qnorm

Concept Check: What would each of the following codes produce?

qnorm( pnorm(3,5,3), 5, 3)

[1] 3

pnorm( qnorm(.2,5,3), 5, 3)

[1] 0.2

Computing probabilities and values

Suppose \(X \sim N(-3.2, 100)\)

that is, normal with expected value -3.2 and variance 100

Probability: What is the probability \(X\) equals 0?

pnorm(0,-3.2,sqrt(100))

[1] 0.6255158

qnorm(.05,-3.2,sqrt(100))

[1] -19.64854

Computing probabilities: SATs

Example

The scores on the SAT math section are \(X \sim N(520,3600)\)

x       = 600
mu      = 520
sigmaSq = 3600
pnorm(x, mu, sqrt(sigmaSq))

[1] 0.9087888

Computing values: SATs

Example

The scores on the SAT math section are \(X \sim N(520,3600)\)

prob    = .96
mu      = 520
sigmaSq = 3600
qnorm(prob, mu, sqrt(sigmaSq))

[1] 625.0412

Transforming Normals: \(aX + b\)

Suppose again \(X \sim N(\mu, \sigma^2)\)

If we multiply/add constants \(a,b\) to \(X\) to form a new rv \(Y\):

\[ Y = aX + b \]

then \(Y \sim N(a\mu + b, a^2 \sigma^2)\)

(This should remind you of our \(E[X]\) and \(Var[X]\) results)

Example:

If \(X \sim N(150,100)\) then

\[ Y = 3X + 30 \sim N(480, 900) \]

Transforming Normals: inches to cm

mu      = 2.54*.875
sigmaSq = 2.54^2*.00005
pnorm(mu + .01, mu, sqrt(sigmaSq)) - 
    pnorm(mu - .01, mu, sqrt(sigmaSq))

[1] 0.4223202

The Standard Normal Distribution

When \(\mu=0\) and \(\sigma=1\), we call it the standard normal

\[ f(x)=\frac1{\sqrt{2\pi}}e^{-x^2/2},\,\,-\infty<x<\infty. \]

(Once again, you don’t need to know this formula for this class)

The standard normal will be notated as: \(Z \sim N(0,1)\)

Standardization

Going from a general normal to a standard normal is known as standardization

\[ X \sim N(\mu, \sigma^2) \longrightarrow Z = \frac{X - \mu}{\sigma} \sim N(0,1) \]

The opposite (actually, the inverse) is true

\[ Z \sim N(0, 1) \longrightarrow X = Z \sigma + \mu \sim N(\mu,\sigma^2) \]

(this is using the results in Transforming Normals)

Z-scores: Comparing values

The z-score can be computed:

\[ \text{z-score} = \frac{x - \mu}{\sigma} \]

zScoreM = (9.63 - 9.85)/sqrt(.0057)
zScoreM

[1] -2.913971

zScoreW = (10.61 - 10.83)/sqrt(.0049)
zScoreW

[1] -3.142857

Z-scores: Another application

Another use of z-scores: what value \(x\) would be equally as unusual?

Answer: Choose the \(x\) so that the z-scores are equal

x = zScoreW * sqrt(.0057) + 9.85
x

[1] 9.612719

Empirical rule

Values from normal distributions contain predictable amounts of probability

This is the empirical rule:

We’ll be back…

The normal plays a central role in probability and statistics

We will return to it later during sampling distributions…